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Theorem xpdom2g 6481
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpdom2g  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )

Proof of Theorem xpdom2g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xpeq1 4418 . . . . 5  |-  ( x  =  C  ->  (
x  X.  A )  =  ( C  X.  A ) )
2 xpeq1 4418 . . . . 5  |-  ( x  =  C  ->  (
x  X.  B )  =  ( C  X.  B ) )
31, 2breq12d 3827 . . . 4  |-  ( x  =  C  ->  (
( x  X.  A
)  ~<_  ( x  X.  B )  <->  ( C  X.  A )  ~<_  ( C  X.  B ) ) )
43imbi2d 228 . . 3  |-  ( x  =  C  ->  (
( A  ~<_  B  -> 
( x  X.  A
)  ~<_  ( x  X.  B ) )  <->  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) ) ) )
5 vex 2617 . . . 4  |-  x  e. 
_V
65xpdom2 6480 . . 3  |-  ( A  ~<_  B  ->  ( x  X.  A )  ~<_  ( x  X.  B ) )
74, 6vtoclg 2671 . 2  |-  ( C  e.  V  ->  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) ) )
87imp 122 1  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436   class class class wbr 3814    X. cxp 4402    ~<_ cdom 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fv 4980  df-dom 6392
This theorem is referenced by:  xpdom1g  6482
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